He arrived today.
Could we call adverb today as an adverbial adjunct because it still complete the meaning of sentence without it?
The term adverb refers to a lexical category while the term adverbial adjunct denotes a syntactic role. The difference is that between categories and (syntactic) functions (which is elaborated in detail in ch.1 of the bulk volume the Cambridge Grammar of the English Language)
If you wanted, you can call an adverb an adverbial, and describe it as an adjunct that provides additional information to the sentence. Oxford Dictionaries explains the respective terms in ways that allow this analysis. (Adverbs are also considered adverbials; as adjuncts they provide extra information,) The result is that it treats a very similar word (yesterday) in a similar sentence (She visited ... yesterday) as an adverbial adjunct:
Adverbial adjuncts can provide extra information about: (...)
when things happen: (...)
I can’t sleep at night.
She visited her family yesterday.
So yes, you can call today, like yesterday in the previous example, an adverbial adjunct.
There are also other explanations for what's going on here. Some guides like The Cambridge Grammar of the English Language (CGEL) analyze today, yesterday, and related words as pronouns (link is to StackExchange answer on subject), and would describe this as a noun phrase used as an adjunct of temporal location. The result is similar to calling it an "adverbial adjunct," but CGEL goes to such lengths to better account for cases where today is clearly not functioning as an adverb.